Representing Measures and Hardy Spaces for the Infinite Polydisk Algebra

Representing measures and Jensen measures are studied for the uniform algebra on the infinite polydiskΔ ¯ ∞generated by the coordinate functions z 1 z 2 ,…. Let σ be the Haar measure on the infinite torus T ∞, which is the distinguished boundary of the infinite polydisk. For fixed p in the range 1 < p , < ∞, it is shown that a point ζ ∈ Δ∞ has a representing measure in L P (σ) if and only if ζ ∈ l 2 . A related result for a class of representing measures for the origin, including the Haar measure σ, and for fixed p in the range 0 < p < ∞, is that the point evaluation at ζ is continuous in the L p ‐norm if and only if ζ ∈ Δ∞ ∩ l 2 . In this case the functions in H p are shown to correspond to analytic functions on the domain Δ∞ l 2 in l 2 . Along these same lines, it is shown that H ∞ (σ) is isometrically isomorphic to H (Δ∞ ∩ l 2 ), and also to H ∞( B ), where B is the open unit ball of the sequence space c 0 .